## What Is Normal Score Transformation?

- Let's study, what is the normal score transformation? How to conduct normal score transformation? How to use normal score transform the data? How to conduct back normal score transformation? Most of the data are not standard normal distribution. The normal score transformation, NST, is designed to transform the dataset, so that it closely resembles a standard normal distribution. Normal distribution equation is shown here. Mu is the mean. Sigma is the standard deviation. It is the function of e powered to x minus the mean squared. Standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal, random variable has a mean of zero, and a standard deviation of one, which is shown in the equation on the graph. Y is the function of e powered to x squared. The graph of the normal distribution depends on two factors: The mean and standard deviation. The mean and the distribution determines the location of the center of the graph, and the standard deviation determines the height of the graph. When the standard deviation is large, the curve is short and wide. When the standard deviation small, the curve is tall and narrow. All normal distributions look like a symmetrical bell-shaped curve. Step of normal score transformation are as the following: The dataset is sorted and ranked from lowest to highest. An equivalent rank from a standard normal distribution is found for each rank from the dataset, and the normal distribution values associated with those ranks make up the transformed dataset. The ranking process can be done using the frequency distribution or the cumulative distribution of the datasets. Examples showing histograms and cumulative distributions before and after a normal score transformation was applied are shown below. Top left figure is the histogram of our original data. Lower left figure is the cumulative frequency of original data, which is used for transforming. Top right figure is the standard normal distribution. Lower left figure is the cumulative standard normal distribution. To transform a porosity value, for example, 0.08, first get the cumulative frequency for porosity equal to 0.08. Go to the same cumulative frequency on standard normal distribution curve, and read the normal score value corresponding to this cumulative frequency value. Here, it is -1.3. Record these data into the table. This is the first paired data of the original and the score. Do the similar things for the second data. Start from Porosity 0.12, get the score -0.09. Put the second pair data into the table. We can finish pairing for all input data. How is Normal Scores used in Sequential Gaussian Simulation? Normal scores shown in the last data table are input to Sequential Gaussian Simulation. Kriging is used to estimate a mean and variance at unsimulated node based on the surrounding input and simulated data and variogram. Local conditional probability distribution, showed as "lcpd," is established by normal distribution and kriged mean and variance. A random value is selected from the local conditional probability distribution and is set to the unsimulated node. Repeat kriging and drawing data, until all grid nodes are populated with standard normal distributed data. To back transform a value, first get the cumulative frequency for this value on standard normal distribution curve, then, go to the same cumulative frequency on original data curve, and read the value corresponding to this cumulative frequency value. For this case, score 0.7 is back transformed to porosity 0.15. Let's review what we have covered. Normal score transformation is to transform the dataset so that it closely resembles a standard normal distribution. To transform a value into normal scores, first get the cumulative frequency for this value, then, go to the same cumulative frequency on standard normal distribution curve, and read the normal score values corresponding to this cumulative frequency value. Normal score transformed data are used in sequential Gaussian simulation. To back transform a value, first get the cumulative frequency for this value on standard normal distribution curve, then, go to the same cumulative frequency on original data curve, and read the value corresponding to this cumulative frequency value. This is the end of the presentation of normal score transformation.

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